The existence of a unique solution to the semi-discrete
problem (REF ) follows from the standard ODE theory in
finite dimensions, in the spirit of Section 9.2 of [1]}.
Here, we only sketch the main steps of the proof.
First, we eliminate the pressure variable and
equation \((\ref {eq:VEM:weak:D})\)  by restricting the (test and trial)
velocity space to divergence-free functions.
Next, () at the initial time \(t=0\)  together with
the definition of \(\mathbf {j}_{h}\)  allows us to define an initial condition
\(\mathbf {E}_{h,0}\)  for the electric field, depending on the initial
conditions \(\mathbf {u}_{h,0}\)  and \(\mathbf {B}_{h,0}\)  of the velocity and
magnetic fields.
Then, we can substitute \((\ref {eq:VEM:weak:B})\)  with the same equation
derived in time; upon further applying (), we
obtain
\(\big [{\mathbf {E}}_{h,t},\mathbf {F}_{h}\big ]_{\operatorname{edge}} + Re_m^{-1} \big [\operatorname{\mathbf {curl}}\mathbf {E}_{h},\operatorname{\mathbf {curl}}\mathbf {F}_{h}\big ]_{\operatorname{face}}+ \big [\partial _t {\mathbf {\chi }}_{h}(\mathbf {v}_{h}, \mathbf {B}_{h}),\mathbf {F}_{h}\big ]_{\operatorname{edge}}= 0\) 
